PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Analog and Digital Filters 575

R.6.56 Observe also that the two parameters that completely specify the Butterworth fi lter

are

The 3 dB cutoff frequency wp

The order n

The Butterworth fi lter uses the Taylor series approximation to model the ideal

LPF prototypes for the two key frequencies: w = 0 and w = ∞ (for any order n).

R.6.57 Analog-Chebyshev fi lters trade off fl atness in the pass band for a shorter transi-

tion band. The main objective of this fi lter is to minimize the design error, where

the error is defi ned as the difference between the ideal LPF (brick wall) fi lter and

the implementation of the actual response of the fi lter over a prescribed band of

frequencies.

The magnitude is either given by an equiripple gain in the pass band and a

monotic behavior in the stop band or a monotonic behavior in the pass band and

equiripple in the stop band, depending on the fi lter type.

Chebyshev fi lters are classifi ed according to their ripple and monotonicity as

Ty p e 1

Ty p e 2

R.6.58 The type-1 Chebyshev transfer function minimizes the difference between the

ideal (brick wall) and the actual frequency response over the entire pass band by

returning equal magnitude ripple in the pass band and a smooth decrease with

maximum fl atness gain in the stop band (see Figure 6.14).

R.6.59 The type-2 Chebyshev fi lter transfer function, also referred to as the inverse Cheby-

shev transfer function fi lter minimizes the difference between the ideal (brick wall)

and the actual frequency response over the entire stop band by returning an equal

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•

•

•

FIGURE 6.14

(See color insert following page 374.) Magnitude plots of normalized analog Chebyshev type-1 LPFs of orders

n = 1, 2, 3, 4, and 10.

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.2

0.4

0.6

0.8

1

Normalized frequency

Gain

Chebyshev type-1 LPF/order n = 1, 2, 3, 4, 10

n = 1

n = 4 n = 3

n = 10

n = 2